The International CAPM Redux Francesca Brusa (Oxford) Tarun Ramadorai (Oxford and CEPR) Adrien Verdelhan (MIT Sloan and NBER) March 2015 Aggregate Foreign Equity Holdings (as % of global GDP) 0.35 Assets As a Percentage of World GDP 0.3 0.25 0.2 0.15 0.1 0.05 0 1970 1975 1980 1985 1990 Portfolios of Foreign Equity Assets 1995 2000 2005 2010 De Jure Index of Financial Openness Motivation I I Large increase in financial integration over the past 30 years. I Aggregate foreign equity holdings, as a % of global GDP: from 3% in the 1980s to approximately 30% in 2011. I Foreign equity holdings of the U.S.: $6 trillion. What are the expected returns on those foreign equity holdings? I Finance logic: I I How to measure bad times? I I If an asset’s return tends to be low in bad times, the asset is risky. Its expected return should be above the risk-free rate. Capital Asset Pricing Model (CAPM): Aggregate market return How to compare returns? I Foreign equity returns are in foreign currencies Currency Risk I Exchange rate risk should matter. . . I I Yet, empirical evidence is difficult to obtain. I I When investors invest abroad, but consume at home, and domestic and foreign purchasing powers differ. International CAPM of Dumas and Solnik (1995): world stock market return + three bilateral exchange rates (Yen, Mark, and Pound). Recent findings: Two risk factors account for a large share of systematic changes in bilateral exchange rates I Idea: Instead of using raw bilateral exchange rates, use currency risk factors that summarize their systematic variation. This Paper 1. Three factors describe the time-series of foreign equity benchmark returns I Smaller difference between expected and realized returns than in the competitors’ models (World CAPM, International CAPM, and Fama-French factors) 2. Those three factors account for a large share of foreign equity mutual funds’ and macro/emerging hedge funds’ returns 3. A simple reduced-form model to study the interaction between currency and equity risk Realized vs Predicted Excess Returns Actual mean excess return (%) World CAPM International CAPM 3 3 2 2 1 1 0 0 −1 −1 0 1 2 −1 −1 3 Actual mean excess return (%) Fama−French Four Factors 3 2 2 1 1 0 0 Aggr. Market 0 1 2 Predicted excess return (%) Value Growth 1 2 3 CAPM Redux 3 −1 −1 0 −1 −1 3 Small Big 0 1 2 Predicted excess return (%) Carry Portfolios 3 Dollar Portfolios Literature I World CAPM: I I Sharpe (1964) and Lintner (1965) to Stehle (1977), Solnik (1974), . . . Korajczyk and Viallet (1989), Harvey (1991), Chan, Karolyi and Stulz (1992), Bekaert and Harvey (1995), Karolyi and Stulz (1996) I International CAPM: I I Solnik (1974), Sercu (1980), Stulz (1981), Adler and Dumas (1983), . . . Bekaert and Hodrick (1992), Ferson and Harvey (1993), Ferson and Harvey (1994), Bekaert (1995, 1996), Dumas and Solnik (1995), De Santis and Gerard (1998), Harvey, Solnik, and Zhou (2002) I New equity factors and frictions: I Chaieb and Errunza (2007), Bekaert, Harvey and Lundblad (2007); Hou, Karolyi, and Kho (2011), Karolyi, Lee and van Dijk (2012), Karolyi and Wu (2014), Malkhozov, Mueller, Vedolin, and Venter (2014) I Currency drivers: I order flows (e.g., Evans and Lyons, 2002, and Froot and Ramadorai, 2005) and risk factors (e.g., Lustig, Roussanov and Verdelhan, 2011, 2014, Menkhoff, Sarno, Scheming, and Schrimpf, 2012, Maggiori, 2012, Lettau, Maggiori, and Weber, 2013, and Verdelhan, 2014) Road map I I I Empirical evidence: I Equity benchmark indices I Hedge funds and mutual funds Theoretical framework: I Intuition I Replication in a reduced-form model Conclusion Empirical Asset Pricing Asset Pricing I When the law of one price holds and investors can form portfolios freely, there exists a SDF Mt+1 that prices any i return Rt+1 : i Et Mt+1 Rt+1 =1 I The same condition holds for the risk-free rate Rf . I Assuming that the returns and SDF are lognormal, the Euler equation implies: i ) covt (mt+1 , rt+1 1 i i ) =− vart (mt+1 ) Et rt+1 − rf ,t + vart (rt+1 2 vart (mt+1 ) | {z } | {z } Λt βti where lower letters denote logs. I What is the SDF Mt+1 ? I If M is an excess return, its market price of risk Λ should be equal to its mean. Equity Literature Summary I World CAPM: WMKTt+1 = world stock market return in U.S. dollars e,i,$ ri,t+1 − rf ,t I i αi + βWMKT [WMKTt+1 − rf ,t ] + t+1 International CAPM: + 3 bilateral exchange rates e,i,$ ri,t+1 − rf ,t I = = i αi + βWMKT [WMKTt+1 − rf ,t ] + i GBP i JPY i EUR βGBP rt+1 + βJPY rt+1 + βEUR rt+1 + t+1 Three/four Fama-French factor models: size, value, and momentum anomalies e,i,$ ri,t+1 − rf ,t = i αi + βWMKT [WMKTt+1 − rf ,t+1 ] + i i i βSMB SMBt+1 + βHML HMLt+1 + βWML WMLt+1 + t+1 Currency Literature Summary I Carry trade risk: sort currencies by their short-term interest rates E (R e ) 1 2 -3.17 -0.46 Portfolios 3 4 1.26 1.80 5 6 2.29 4.48 Source: Lustig, Roussanov, and Verdelhan, 2011. I “Dollar” risk: sort currencies by their exchange rate betas, long if average interest rate > U.S. interest rate, short otherwise E (R e ) 1 2 1.31 2.86 Portfolios 3 4 3.87 5.13 Source: Verdelhan, 2014. 5 6 4.99 7.14 Intuition I Three risk factors: Equity, Dollar, and Carry. 1 i i Et rt+1 − rf ,t + vart (rt+1 ) = βti,1 Λ1t + βti,2 Λ2t + βti,3 Λ3t . 2 I One could ignore currency risk and focus on equity risk only if all the quantities and prices of risk were in sync. 1 f i i Et rt+1 − rf ,t + vart (rt+1 ) = βti, Λet . 2 I No reason to expect so. This Paper: International CAPM Redux I Three-factor model: e,i,$ Carry i i Dollar i ri,t+1 − rf ,t = αi + βLWMKT [LWMKTt+1 − rf ,t ] + βDollar rt+1 + βCarry rt+1 + t+1 I Factors: 1. Global equity (LWMKT): built from local currency equity returns. 2. Dollar: average excess return of a U.S. investor going long foreign Treasury Bills. 3. Carry: average excess return of a U.S. investor going long high interest rate currencies and short low interest rate currencies. I Estimation: time-varying betas obtained on 60-month rolling windows. Data and Estimation I Test assets: from 46 developed and emerging countries, spanning value, growth, and country index returns from 1/1976 to 12/2013. I Risk factors: global equity factor (built from local currency equity returns) and two currency factors: Dollar and Carry I Estimation: time-varying betas obtained on rolling windows Test Assets (I/II) Developed Markets 20 15 % 10 5 0 −5 −10 Portugal Ireland Japan Austria Israel New Zealand Italy Greece United States Spain Canada Germany United Kingdom France Singapore Belgium Countries Switzerland Denmark Netherlands Norway Australia Finland Sweden China Hong Kong Aggr. Market Growth Small Big Value Growth Aggr. Market Value Big Asset type Small Test Assets (II/II) Emerging Markets 30 25 20 % 15 10 5 0 −5 China Morocco Taiwan Malaysia Philippines South Korea India South Africa Czech Republic Thailand Hungary Chile Egypt Colombia Peru Indonesia Countries Poland Mexico Turkey Russia Brazil China Morocco Taiwan Aggr. Market Growth Small Big Value Growth Aggr. Market Value Big Asset type Small Expected vs Realized Equity Excess Returns I Since factors are excess returns, the market prices of risk should be the means of the risk factors (no-arbitrage condition) ET I I e,i,$ rt+1 − rf ,t e,i,$ rt+1 − rf ,t = αi + β i Ft+1 + t+1 = αi + β i ET (Ft+1 ) For each asset and each rolling window: I Estimate the beta on rolling windows, from t − 59 to t: βt I Estimate the price of risk as the mean of the risk factor over the P same period, from t − 59 to t: 1 λt = 60 59 i=0 Ft−i I Form expected equity excess returnsas the product of e,i,$ the quantities and prices of risk: Et rt+1 − rf ,t = βt λt Average across time and compare expected to realized excess returns Time-varying Factor Betas: Developed Markets Exposure to LWMKT Exposure to Dollar Exposure to Carry Australia Austria Belgium Canada Denmark Finland France Germany Greece Hong Kong Ireland Israel Italy Japan Netherlands New Zealand Norway Portugal Singapore Spain Sweden Switzerland United Kingdom United States −1 0 1 βLWMKT 2 3 −1 0 Average 1 βDollar 2 3 Min/Max −1 0 1 βCarry 2 3 Time-varying Factor Betas: Emerging Markets Exposure to LWMKT Exposure to Dollar Exposure to Carry Brazil Chile China Colombia Czech Republic Egypt Hungary India Indonesia Malaysia Mexico Morocco Peru Philippines Poland Russia South Africa South Korea Taiwan Thailand Turkey −4 −2 0 2 βLWMKT 4 6 −4 −2 0 Average βDollar 2 4 6 Min/Max −4 −2 0 βCarry 2 4 6 Realized vs Predicted Excess Returns Actual mean excess return (%) World CAPM International CAPM 3 3 2 2 1 1 0 0 −1 −1 0 1 2 −1 −1 3 Actual mean excess return (%) Fama−French Four Factors 3 2 2 1 1 0 0 Aggr. Market 0 1 2 Predicted excess return (%) Value Growth 1 2 3 CAPM Redux 3 −1 −1 0 −1 −1 3 Small Big 0 1 2 Predicted excess return (%) Carry Portfolios 3 Dollar Portfolios Robustness Checks I Other comparisons of the pricing errors: I Histograms and kernel density estimates of rolling alphas I Rolling mean absolute alphas I Even if the market prices of risk are estimated (Fama-McBeth procedure), the CAPM Redux still compares favorably: I the market prices of risk are positive and significant for our three factors I equity risk is priced for the CAPM, International CAPM, and FF models, but not the other factors (except GBP); the prices of risk appear further removed from the mean of the risk factors. I Subsamples: I time-windows (increasing sizes starting from 1976 or backwards from 2013) I test assets (country aggregates, developed vs emerging markets) I length of the rolling windows Kernel Density of Rolling Alphas 0.6 Kernel density 0.4 0.2 0 −5 −3 −1 World CAPM 1 3 Rolling alphas Fama−French Four Factors 5 7 CAPM Redux 9 Rolling Mean Absolute Alphas 1.6 1.4 Mean Absolute Alphas 1.2 1 0.8 0.6 0.4 0.2 Jan−81 Aug−85 WCAPM Mar−90 Oct−94 Jun−99 Jan−04 Fama−French Four Factors Aug−08 Apr−13 CAPM Redux Correlation Between Equity Returns & Currency Factors LOC Equity Returns and Carry factor 1 0.8 0.8 0.6 0.6 0.4 0.2 Corr(Ret, Carry)t Corr(Ret, Dollar)t LOC Equity Returns and Dollar factor 1 0.4 0.2 0 0 −0.2 −0.2 −0.4 −0.4 Jan−81 Aug−85 Apr−90 Nov−94 Jun−99 Jan−04 Sep−08 Apr−13 Jan−81 Aug−85 Apr−90 Nov−94 Jun−99 Jan−04 Sep−08 Apr−13 Mutual and Hedge Fund Returns Hedge Funds and Mutual Funds I Fund’s exposure to global systematic factors: i Rt+1 = αi + β i Carryt+1 + γ i Dollart+1 + δ i LWMKTt+1 + εit+1 , i where Rt+1 denotes the monthly realized return of fund i. I Data: I MFs: U.S. “Foreign Equity Funds” from CRSP. I HFs: International “Macro” and “Emerging” Funds from Ramadorai (2013) and Patton, Ramadorai and Streatfield (2013). I Estimation: time-varying betas obtained on rolling windows. I Sample period: November 1990 (MF)/January 1994 (HF) to April 2013. Mutual Funds’ Significant Exposure to Currency Factors Number of funds 1200 600 0 1996 1998 2000 2002 2004 2006 2008 2010 2012 1996 1998 2000 2002 2004 2006 2008 2010 2012 1996 1998 2000 2002 2004 2006 2008 2010 2012 1 Dollar Factor 0.8 0.6 0.4 0.2 0 1 Carry Factor 0.8 0.6 0.4 0.2 0 “Macro/Emerging” Hedge Funds’ Significant Exposure to Currency Factors Number of funds 400 200 0 2000 2002 2004 2006 2008 2010 2012 2000 2002 2004 2006 2008 2010 2012 2000 2002 2004 2006 2008 2010 2012 1 Dollar Factor 0.8 0.6 0.4 0.2 0 1 Carry Factor 0.8 0.6 0.4 0.2 0 Statistical Significance of Positive HF and MF’s Alphas Hedge Funds 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 ECDF ECDF Mutual Funds 1 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 1 2 3 4 5 0 1 t(α>0) World CAPM 2 3 t(α>0) Int. CAPM Fama−French Four Factors CAPM Redux 4 5 Theoretical Framework Intuition I Recall that: Et i rt+1 − rf ,t + i ) covt (mt+1 , rt+1 1 i vart (rt+1 ) =− vart (mt+1 ) 2 vart (mt+1 ) | {z } | {z } Λt βti I Expected currency excess returns when markets are complete and shocks are gaussian (Bekaert, 1996, Bansal, 1997, Backus, Foresi, and Telmer, 2001): i Et rxt+1 = i rfi ,t − rf ,t − Et ∆st+1 = 1 1 i Vart (mt+1 ) − Vart (mt+1 ) 2 2 I Failure of the uncovered interest rate parity (U.I.P.) implies that expected currency excess returns are time-varying. I SDF must be heteroscedastic, and thus market prices of risk must be time-varying. Framework I Assume that financial markets are complete (thus log changes in exchange rates are differences in log SDF) I Start from the law of motion of each SDF, which depends on country-specific and global shocks I Assume that each country-aggregate dividend growth rate depends on the same shocks as the SDF I Two key assumptions: I Prices of risk are time-varying (time-varying currency risk premia) I One global shock affects all SDF in the same way (role for a pure equity risk factor) Reduced-Form Model with Endogenous Exchange Rates I In the tradition of Frachot (1996), Backus, Foresi and Telmer (2001), Brennan and Xia (2006) I Exponentially affine SDFs in complete financial markets i −mt+1 = + i zt+1 = w zt+1 = i ∆dt+1 = + q i α + χzti + γzti ut+1 q q p g w c τ ztw + δ i ztw ut+1 + κzti ut+1 , + ωztw ut+1 q i (1 − φ)θ + φzti − σ zti ut+1 , p w w w w w w (1 − φ )θ + φ zt − σ ztw ut+1 q i µD + ψzti + ψw ztw + σD zti ut+1 q p g g c,i p w c w w σD ztw ut+1 + σD zti ut+1 + σD zt ut+1 i w , ug , uc where ut+1 , ut+1 t+1 t+1 are i.i.d, mean-zero, variance-one Gaussian shocks. I Similar model studied in Lustig, Roussanov, and Verdelhan (2011, 2014) and in Verdelhan (2014) but without the global “equity” c shocks ut+1 Model Solution I Simple and tractable reduced-form model: I I I Interest rates, exchange rates, and price-dividend ratios in closed-forms, as well as realized and expected equity and currency excess returns, and equity and currency risk factors. Implications: I The world stock market return in U.S. dollars depends on all global shocks. I No role for bilateral exchange rates in theory I But in the model the market price of global equity risk depends on several state variables. I Those state variables are unknown to the econometrician. I Quantities and market prices of risk evolve at different frequencies. Role for currency factors in practice, even in simulated data Realized vs Predicted Excess Returns: Simulated Data World CAPM CAPM Redux 7 Realized average excess returns (%) Realized average excess returns (%) 7 6.5 6 5.5 5 4.5 4 3.5 3 6.5 6 5.5 5 4.5 4 3.5 3 3 3.5 4 4.5 5 5.5 6 6.5 7 3 3.5 4 4.5 5 5.5 6 6.5 7 Predicted excess returns (%) Predicted excess returns (%) Conclusion I Three factors (Global Equity, Dollar, and Carry) describe the time-series of foreign equity benchmark returns. I Predicted and realized equity excess returns are closer than in the competitors’ models. I Currency risk matters for foreign equity mutual fund and “macro/emerging” hedge fund returns. I Interaction between currency and equity risk in a simple reduced-form model: I Time-variation in quantities and prices of risk is key.